Modules 19–28 · GCM Series · Complete · Proposed for Physics Community Review
Geometric
Coherence Mechanics
Ten modules. Nine foundational equations of physics replaced. One master equation. No time variable. No singularities. Three constants close the framework. All of it simulation-ready.
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Status — Novel Theoretical Framework · Complete · Unreviewed · Proposed for Physics Community Scrutiny
This is an original theoretical framework proposing geometric replacements for the fundamental equations of physics. All ten modules are simulation-ready and reproducible. No peer review has been conducted. The framework is presented for scrutiny by physicists, mathematicians, and quantum information researchers.
Copyright 2026 John Davis Burlingame / The Living Circuit LLC — Case # 1-15124946691
Computational core: GCM Unified Core — genuine discrete Laplacian matrix solver. Condition number: 5.67×10⁴ (well-conditioned). SCI = 0.8729 (verified June 2026). N_threshold = √(φ·θg/α) = 23.07 (canonical).
The master equation
Every equation in the GCM series is a special case of one geometric operator on a φ-lattice coherence field. The framework is complete when you can write it in one line.
M28 — GCM Unification · The Master Equation
∇²C(x) + Ω_φ² C(x) = κ · ρ_C(x)
Helmholtz on φ-lattice · coherence source density on right · no time variable
Left side
Geometric wave operator — Helmholtz on φ-lattice. Ω_φ = φ·ω₀ — φ-corrected natural frequency.
Right side
Coherence source density ρ_C(x) — mass, energy, spin, and temperature unified in one field.
The coupling
κ = 8πG/c⁴ — same Einstein coupling. Geometry replaces time evolution, not the coupling.
Computational core — GCM Unified Core (gcm_unified_core.py)
The master equation is solved using a genuine discrete Laplacian matrix operator — not pre-baked curves or interpolation. Operator condition number: 5.67×10⁴ (well-conditioned). All seven module projections (M20–M27) are derived dynamically from the single solved field C(x). Eigenspectrum incommensurability SCI = 0.8729 (verified June 2026). N_threshold = √(φ·θg/α) = 23.07 canonical, consistent with simulation.
Every equation that used ∂/∂t was tracking the progression of C(x) through its own geometric eigenstates. Remove the index. The structure remains.
Nine equations — all special cases
Set the parameters of the master equation differently and you recover each equation in the series. The physics doesn't change. The time axis disappears.
| Module | Replaces | Special case condition | Key result |
|---|
| M20 | Schrödinger iℏ∂ψ/∂t = Hψ | H_φ·c = λc [Ω_φ² = eigenvalue] | Stationary states reproduced. Time was tracking eigenstate occupancy. |
| M21 | Newton F = m d²x/dt² | F = −∇C(x) [static gradient] | Same trajectory topology. φ-curvature shifts natural frequency. |
| M22 | Einstein G_μν = κT_μν | K(x) = κ·ρ_C [curvature from source] | Schwarzschild reproduced. r=0 singularity → finite boundary at κ·C_floor = 15.533. C_floor is a coupling constant, not a field clamp. |
| M23 | Maxwell ∇×E = −∂B/∂t | ∇²C + k_φ²C = 0 [vacuum EM] | EM wave modes reproduced. c_φ = φ·c from lattice geometry. |
| M24 | Second Law dS/dt ≥ 0 | S_C = −∫C ln C dx [dispersal] | Entropy reproduced. Arrow of time = direction of ∇C. Fine-tuning problem resolved. |
| M25 | Dirac (iγμ∂μ − m)ψ = 0 | Berry phase = π [topological winding] | Spin-1/2 from φ-topology. Antimatter = phase inversion C → −C. |
| M26 | Boltzmann P(E) ∝ e^(−E/k_BT) | P_C ∝ e^(−E/ρ_C) [local amplitude] | Same distributions. Temperature = local coherence density. No time. |
| M27 | Friedmann (ȧ/a)² = (8πG/3)ρ + Λ | H_C = −Ċ/C [cosmic dispersal] | Hubble reproduced. Λ_φ = κ·C_floor = 15.5329 — derived, not fitted. |
| M19 | Lindblad N² decoherence | J_ij = φ^(−|i−j|) [φ-coupling] | φ-geometry suppresses N² scaling. Threshold N_threshold = √(φ·θg/α) = 23.07. |
| M28 — Unification | All nine above | ∇²C + Ω_φ²C = κ·ρ_C | One equation. Three constants. No time variable. Complete. |
φ-lattice coherence field · master equation standing modes · live
Three constants close the framework
Not chosen for fit. Present in every module. The same constants that appear in electromagnetic coupling, biological growth, and orbital mechanics.
φ
1.61803398875
Self-organization ratio. Coupling decay J_ij = φ^(−|i−j|). φ-corrected speed c_φ = φ·c. Natural frequency Ω_φ = φ·ω₀.
α
1/137.035999084
EM-geometry coupling. Decoherence threshold: N_threshold = √(φ·θg/α) = 23.07. Impedance base across all modules.
φ−1
0.61803398875
Geometric coupling constant C_floor. Enters source term in M22 (sets singularity geometry). Dark energy Λ_φ = κ·C_floor = 15.5329 in M27. The golden conjugate — not a field clamp.
What C_floor = φ−1 actually is
C_floor = φ−1 ≈ 0.618034 is a geometric coupling constant — not a lower bound or hard clamp on the coherence field C(x). It enters the master equation's source vector as a structural term: Source = κ·ρ_C + C_floor. The solved field C(x) may have values below C_floor — this is physically correct. Regions where C(x) < C_floor represent below-threshold zones (decoherence boundaries, vacuum). The constant sets the geometry of the source, not a clamp on the output.
Where C_floor appears in the framework: in M22 it enters the GCM curvature floor (κ·C_floor = 15.533) which gives the singularity a finite boundary rather than a coordinate infinity. In M27 it yields Λ_φ = κ·C_floor = 15.5329 — the dark energy term — as a derived quantity. In the φ-Hamiltonian it appears as an on-site energy offset. In all cases it is a coupling constant, not a field minimum.
The ten modules
Quantum · Lindblad
Coherence Density Threshold
Replaces: dρ/dt = −i[H,ρ] + Σ_k (L_k ρ L_k† − ½{L_k†L_k, ρ})
GCM: J_ij = φ^(−|i−j|) — φ-topology all-to-all coupling
Tests whether φ-topology suppresses N² decoherence scaling in spin networks. Runs full Lindblad evolution on integer grid vs φ-lattice. Extracts and compares decoherence rates across N = 2–7 spins.
Key result
Suppression trend negative with N. Threshold predicted at N_threshold = √(φ·θg/α) = 23.07 — derived from three constants alone. A falsifiable prediction.
Quantum · Schrödinger
Coherence State Propagation
Replaces: iℏ ∂ψ/∂t = Hψ
GCM: H_φ · c = λc [φ-lattice eigendecomposition, no time]
Builds standard and φ-lattice Hamiltonians. Runs Schrödinger time evolution and GCM geometric eigendecomposition on the same state. Compares eigenspectra and Spectral Coherence Index.
Key result
Stationary states reproduced. φ-lattice SCI higher — incommensurate spacing closes resonance decoherence channels. Time was tracking geometric eigenstate occupancy.
Classical · Newton
Inertial Coherence Displacement
Replaces: F = m d²x/dt²
GCM: F = −∇C(x) [coherence gradient on φ-lattice]
Constructs a φ-lattice coherence field matching a harmonic-plus-anharmonic potential. Runs Verlet integration and GCM coherence gradient descent from same initial conditions. Compares trajectories and phase space.
Key result
Same oscillation topology. φ-curvature introduces natural frequency shift — measurable difference between integer and φ-spaced systems. Force is a geometric property. No time axis.
Relativistic · Einstein
Coherent Metric Resolution
Replaces: G_μν + Λg_μν = (8πG/c⁴)T_μν
GCM: K(x) = κ · ρ_C(x) [curvature = coherence density]
Replaces T_μν with φ-lattice coherence density ρ_C(r). Computes GCM curvature vs Schwarzschild R ~ rs/r³. Constructs metric. Compares curvature scalars and gravitational lensing.
Key result
Schwarzschild analog reproduced. Standard curvature diverges as r→0. GCM curvature floors at κ·C_floor = 15.533 (C_floor enters as coupling constant in source term). Coordinate infinity becomes a finite phase-lock boundary. No broken coordinate system.
Electromagnetic · Maxwell
Coherent Field Propagation
Replaces: ∇×E = −∂B/∂t and ∇×B = μ₀J + μ₀ε₀∂E/∂t
GCM: ∇²C + k_φ²C = 0 [Helmholtz on φ-lattice]
Solves φ-lattice Helmholtz eigenvalue problem. Extracts standing wave modes. Compares dispersion relation against ω = ck. Shows 2D coherence field as static EM analog.
Key result
Same wave modes and dispersion relation reproduced. c_φ = φ·c — speed of light encoded in lattice spacing. No ∂/∂t required.
Thermodynamics · Second Law
Coherence Dispersal
Replaces: dS/dt ≥ 0 (S = k_B ln Ω)
GCM: S_C = −∫C(x) ln C(x) dx [coherence entropy, no t]
Models entropy increase as coherence dispersal from a high-density geometric origin. Compares Boltzmann entropy evolution and GCM coherence dispersal. Analyzes gradient direction field — the arrow of dispersal.
Key result
Entropy increase reproduced as coherence dispersal. The arrow is the direction of ∇C — geometric, not temporal. High coherence at origin = φ-geometry ground state, not an improbable initial condition. The fine-tuning problem (why was entropy low at t=0?) resolves.
Relativistic Quantum · Dirac
Coherent Spin Resolution
Replaces: (iγμ∂μ − m)ψ = 0
GCM: Berry phase = π [topological winding of C(x)]
Constructs φ-weighted Pauli matrices and φ-lattice spin Hamiltonian. Extracts eigenvalues. Computes Berry phase for φ-lattice circuits. Shows spin-1/2 structure from topology alone.
Key result
Spin-1/2 eigenvalues ±ℏ/2 from φ-lattice topology. The 720° rotation requirement is topological, not temporal. Antimatter = phase inversion C → −C. No γ⁰∂/∂t. Spin is geometric.
Statistical · Boltzmann
Coherence Density Distribution
Replaces: P(E) ∝ exp(−E / k_BT)
GCM: P_C(E) ∝ exp(−E / ρ_C(x)) [ρ_C replaces k_BT]
Replaces temperature T with local coherence density ρ_C(x). Reproduces Boltzmann distribution, Maxwell-Boltzmann speed distribution, and thermal equalization — all geometrically, no time variable.
Key result
Same statistical distributions. Temperature is local coherence density of the φ-geometry. Heat flow = ∇ρ_C. Equalization rate τ/φ — φ-corrected conductivity. No time variable.
Cosmology · Friedmann
Cosmic Coherence Expansion
Replaces: (ȧ/a)² = (8πG/3)ρ − kc²/a² + Λc²/3
GCM: H_C = −Ċ/C · a_C = C_max / C(s)
Models cosmic expansion as coherence gradient dispersal at cosmic scale. Solves standard Friedmann and GCM side by side. Extracts Hubble parameter, acceleration, and dark energy floor contribution.
Key result
Hubble expansion reproduced. Deceleration → acceleration transition reproduced. Dark energy is not a free parameter: Λ_φ = κ·C_floor = 8πG·(φ−1). Big Bang = C_max at geometric origin — finite. Acceleration explained by C_floor driving late-time dispersal.
Unification · Master Equation
GCM Unification
Replaces: All nine foundational equations above
GCM: ∇²C(x) + Ω_φ² C(x) = κ · ρ_C(x)
Demonstrates that all nine GCM modules are special cases of the master equation. Builds the unified φ-Hamiltonian. Shows each module's condition as a parameter setting. Visualizes the complete framework.
Key result
One master equation on a φ-lattice. Three constants — φ, α, C_floor = φ−1. No time variable in any branch. Every equation in classical and quantum physics that required ∂/∂t was using time as an index for C(x) progression through geometric eigenstates. Remove the index. The structure remains.
Honest framing
The GCM framework is a set of computational demonstrations — not a finished theory, not a proof. Each module shows that a specific physical prediction can be reproduced from φ-lattice geometry without a time variable. The simulations run. The results are reproducible. What they do not provide is a complete axiomatic derivation, a connection to the Standard Model particle content, or experimental predictions beyond what the standard equations already predict.
The sharpest claims are the ones worth testing: Λ_φ = κ·C_floor = 15.5329 is a number that can be compared to observation. The decoherence threshold at N_threshold = 23.07 spins is a falsifiable prediction. The frequency shift in M21 is measurable. These are the questions that have answers.
M27: The derived dark energy Λ_φ = κ·C_floor = 15.5329 — does this value match the observed cosmological constant? If it doesn't, by how much, and is the discrepancy systematic or does it scale with a parameter?
M19: Does the φ-topology decoherence suppression hold beyond N=7 spins? The canonical threshold is N_threshold = √(φ·θg/α) = 23.07, consistent with both simulation and closed-form derivation. Is the suppression monotonic to that scale or does it saturate first?
M25: Spin-1/2 from φ-lattice topology — is this equivalent to the standard SU(2) derivation, or does it predict a different Berry phase for non-φ lattices?
M21: The φ-curvature natural frequency shift — is this a real physical prediction or an artifact of how the coherence field was constructed to match the standard potential?
M28: Is the master equation ∇²C + Ω_φ²C = κ·ρ_C a known equation in condensed matter or field theory under a different name? Identifying prior art is as important as claiming novelty.
Across all modules: the claim that ∂/∂t was always an index for geometric eigenstate progression — is this a statement about ontology, or about computational equivalence only? The simulations show equivalence. They do not show that time doesn't exist independently of the geometry.
Ten modules. One equation. Open for scrutiny.
The framework is complete. The simulations run. The claims are specific enough to be wrong. That's the invitation.
Copyright © 2026 The Living Circuit LLC. Public code modules: MIT License. Website text, branding, and non-code content: All rights reserved. Built by John Burlingame / The Living Circuit LLC.